quiz8solns - STA 4321/5325 — Spring 2010 Quiz 8 — April...

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Unformatted text preview: STA 4321/5325 — Spring 2010 Quiz 8 — April 9 Name: my There are five problems in this quiz. Each problem has exactly one correct answer. Problem 1 Let f (as, y) denote the joint probability density function of two continuous random variables X and Y, and f X(m) denote the marginal probability density function of X. Then it is always true that (a) fx($) = If; H1737de (b) fx(w) = Li” f(m, y)dy (C) fx 2? (d) fx(w)=f:’;f<x,y)dy ' Problem 2 Let X and Y be two discrete random variables with the following joint probability mass function: Y Way) 0 1 2 gix‘x) 0 1/9 2/9 1/9 V777,“ X 1 2/9 2/9 0 577 2 1/90 0 E :9 x H s :1 HI» q; E, \_, :2 a; II II |—‘ Old: {DIN 00 U (b ,‘E x fl ll @ :9 N H I: ll Problem 3 Let X and Y be two continuous random variables with joint probability density function f (ac, y), and marginal density functions fX and fy respectively. Then X and Y are said to be independent if f (2:, y) = fX(£L’)fy(y) for every :3 e R, g E R. This statement is (b) False Problem 4 Let X and Y be continuous random variables taking positive values, with joint probability density function given by f (m, y) = e’(‘”+y) for every .1: > 0, y > 0. It can be derived that the marginal probability density function of Y is given by fy(y) = e‘y for y > 0. Then, the conditional probability density function of X given Y = 7 is given by % C )‘3 I y: C (a) ley:7(m) = 6-7 for every x > 0 fix] _: ’f 7}— ‘lgr L 7) 7 C (b) fX|Y=7($) 2 84$”) for every .7: > 0 z (c) ley:7(:r) = 6—9“ for every x > 0) IX» (d) fX|Y=7(CI7) = e_(y+7) for every (E > 0 : 6 Problem 5 Let X and Y be two independent Bernoulli random variables. Let P(X = 0) : and P(Y = 0) 2 Then, fixed-=1- r-) fix“): Z WIH ll ti)th “Nib (a) P(X =0,Y= 1)- ~ (C) p(X=0’Y:1) ‘ {7(120}:1—-c7 VLYfJ): (d) P(X = O,Y = 1) ©|l-‘ «:10. CLE— 3*3 2;. ...
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quiz8solns - STA 4321/5325 — Spring 2010 Quiz 8 — April...

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